A373272 Triangle read by rows: T(n,k) = sum of all distinct multiplicities in the integer partitions of n with k parts.

1, 1, 2, 1, 1, 3, 1, 3, 3, 4, 1, 2, 6, 4, 5, 1, 4, 7, 6, 5, 6, 1, 3, 10, 11, 10, 6, 7, 1, 5, 11, 16, 14, 12, 7, 8, 1, 4, 15, 20, 22, 14, 14, 8, 9, 1, 6, 16, 26, 28, 29, 20, 16, 9, 10, 1, 5, 20, 34, 41, 40, 34, 23, 18, 10, 11, 1, 7, 22, 42, 50, 54, 44, 35, 26, 20, 11, 12, 1, 6, 26, 52, 69, 75, 68, 54, 44, 29, 22, 12, 13

Offset: 1

Author

Olivier Gérard, May 29 2024

Keywords

• nonn
• tabl

Examples

			Array begins:
  1;
  1, 2;
  1, 1,  3;
  1, 3,  3,  4;
  1, 2,  6,  4,  5;
  1, 4,  7,  6,  5,  6;
  1, 3, 10, 11, 10,  6,  7;
  1, 5, 11, 16, 14, 12,  7,  8;
  1, 4, 15, 20, 22, 14, 14,  8, 9;
  1, 6, 16, 26, 28, 29, 20, 16, 9, 10;
  ...
T(6,3) = 7 because the partitions of 6 into 3 parts are 4+1+1, 3+2+1, 2+2+2,
  the multiplicities are (1,2), (1,1,1), (3),
  the distinct multiplicities are respectively (1,2), (1), (3),
  contributing 3+1+3 = 7.
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened (first 40 rows from Olivier Gérard)

Crossrefs

Columns k=1-2 give: A057427, A028242.

Main diagonal gives A000027.

Row sums are A373273.

T(2n,n) gives A373104.

Programs

Mathematica

Flatten[Table[
  Plus @@@
   Table[Map[Plus @@ Union[Length /@ Split[#]] &,
     IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]]